Question 1

Q1: How many tangents can a circle have?

Accepted Answer

Answer: A circle can have infinitely many tangents. Reason: A circle is a collection of infinitely many points. A unique tangent can be drawn at each of these points. Therefore, a circle can have an infinite number of tangents.

Question 2

Q2: Fill in the blanks : (i) A tangent to a circle intersects it in _____ point (s). (ii) A line intersecting a circle in two points is called a _____ . (...

Accepted Answer

Answers: (i) A tangent to a circle intersects it in one point(s). (ii) A line intersecting a circle in two points is called a secant. (iii) A circle can have two parallel tangents at the most. (These tangents are at the ends of a diameter). (iv) The common point of a tangent to a circle and the circ...

Question 3

Q3: A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is : (A) 12 cm (B) 1...

Accepted Answer

Answer: (D) $\sqrt{119}$ cm. Explanation: Given: Radius of the circle, $OP = 5 	ext{ cm}$. Distance of point Q from the center, $OQ = 12 	ext{ cm}$. PQ is the tangent to the circle at point P. To Find: The length of the tangent, PQ. Solution: According to Theorem 10.1, the tangent at any point of...

Question 4

Q4: Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Accepted Answer

To Construct: A circle, a given line, a tangent parallel to the given line, and a secant parallel to the given line. Steps of Construction: 1.  Draw a circle with center O and any suitable radius. 2.  Draw a line, let us call it L, outside the circle. This is the 'given line'. 3.  From the center O,...

Question 5

Q1: From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B...

Accepted Answer

Answer: (A) 7 cm Explanation: Given: Let the circle have its center at O. Let Q be the external point. Length of the tangent from Q, let us say QT, is $QT = 24 	ext{ cm}$. Distance of Q from the center, $OQ = 25 	ext{ cm}$. To Find: The radius of the circle, $OT$. Solution: The radius at the point...

Question 6

Q2: In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that $\angle POQ = 110^\circ$, then $\angle PTQ$ is equal to (A) $60^\ci...

Accepted Answer

Answer: (B) $70^\circ$ Explanation: Given: TP and TQ are tangents to a circle with center O. $\angle POQ = 110^\circ$. To Find: $\angle PTQ$. Solution: We know that the radius is perpendicular to the tangent at the point of contact. Therefore, $OP \perp TP$, which means $\angle OPT = 90^\circ$. And,...

Question 7

Q3: If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of $80^\circ$, then $\angle POA$ is equal to (A) $5...

Accepted Answer

Answer: (A) $50^\circ$ Explanation: Given: PA and PB are tangents from point P to a circle with center O. The angle between the tangents, $\angle APB = 80^\circ$. To Find: $\angle POA$. Solution: In triangles $	riangle OAP$ and $	riangle OBP$: $OA = OB$ (Radii of the same circle) $PA = PB$ (Length...

Question 8

Q4: Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Accepted Answer

To Prove: The tangents drawn at the ends of a diameter of a circle are parallel. Proof: Let us consider a circle with center O and diameter AB. Let PQ be the tangent to the circle at point A, and let RS be the tangent to the circle at point B. We know that the radius is perpendicular to the tangent...

Question 9

Q5: Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Accepted Answer

To Prove: The perpendicular at the point of contact to the tangent to a circle passes through the centre. Proof: Let us consider a circle with center O. Let XY be a tangent to the circle at a point P. We need to prove that the line perpendicular to XY at P passes through the center O. We know from T...

Question 10

Q6: The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Accepted Answer

Given: Let the circle have its center at O. Point A is at a distance of 5 cm from the center, so $OA = 5 	ext{ cm}$. The length of the tangent from point A is 4 cm. Let P be the point of contact, so $AP = 4 	ext{ cm}$. To Find: The radius of the circle, $OP$. Solution: The radius is perpendicular...

Question 11

Q7: Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Accepted Answer

Given: Two concentric circles with a common center O. Radius of the larger circle, $R = 5 	ext{ cm}$. Radius of the smaller circle, $r = 3 	ext{ cm}$. AB is a chord of the larger circle that is a tangent to the smaller circle at point P. To Find: The length of the chord AB. Solution: Join OA and O...

Question 12

Q8: A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that $AB + CD = AD + BC$

Accepted Answer

To Prove: For a quadrilateral ABCD circumscribing a circle, $AB + CD = AD + BC$. Proof: Let the quadrilateral ABCD touch the circle at points P, Q, R, and S on the sides AB, BC, CD, and DA, respectively. We know that the lengths of tangents drawn from an external point to a circle are equal. Therefo...

Question 13

Q9: In Fig. 10.13, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and...

Accepted Answer

To Prove: $\angle AOB = 90^\circ$. Proof: Let the circle have center O. Let XY be a tangent at point P and X'Y' be a tangent at point Q. Since XY is parallel to X'Y', PQ must be a diameter of the circle. Let AB be another tangent to the circle at point C. Join OA, OB, and OC. Consider triangles $	r...

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